Abstract: In this paper we introduce ``higher extremal Kähler'' metrics. We provide an example of the same on a minimal ruled surface. We also prove a perturbation result that implies that there are non-trivial examples of ``higher constant scalar curvature'' metrics, which are basically metrics where the top Chern form is harmonic. We also give a relatively short proof of Liu's formula for the Bando-Futaki invariants (which are obstructions for the existence of harmonic Chern forms) of hypersurfaces of projective space.